Subgroups — #20283425

Order 6784.

500 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133223	C₂	1	1	yes	maximal, characteristic, complemented
2	#16811169	C₂	1	1	yes	maximal, characteristic
3	#16810699	C₂	1	4	yes	maximal, complemented
4	C₈×C₂²⋊D₅₃	C₂	1	4	yes	maximal, complemented
5	#16810646	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810591	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810590	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#131×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#11719945	C₂²	1	1	yes	characteristic
11	#13133223	—	2	4	no	—
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₂×#11719912	C₂²	1	2	yes	—
14	C₂²×#88322	C₂²	1	1	yes	characteristic
15	C₂×C₈×D₅₃	C₂²	1	2	yes	—
16	#13133215	C₄	1	1	yes	characteristic
17	#13133213	C₄	1	1	yes	characteristic
18	#13133206	C₄	1	1	yes	characteristic
19	#13133204	C₄	1	1	yes	characteristic
20	#13133189	C₄	1	4	yes	—
21	C₄×C₂²⋊D₅₃	C₂²	1	4	yes	—
22	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
23	#13133146	C₂²	1	4	yes	—
24	C₂²⋊#88322	C₂²	1	4	yes	—
25	#13133129	C₂²	1	4	yes	—
26	C₈×Dic₅₃	C₂²	1	2	yes	—
27	C₈×Dic₅₃	C₂²	1	2	yes	—
28	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
29	#131×C₅₃	—	2	4	no	—
30	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
32	C₂²×C₈×C₅₃	C₂²	1	2	yes	—
33	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	#41×C₅₃	—	2	2	no	—
35	#41×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
38	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₂²⋊D₅₃	C₂×C₄	1	4	yes	—
42	C₂×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
43	C₂²×Dic₅₃	C₂×C₄	1	2	yes	—
44	#11719945	—	2	2	no	—
45	#11719945	—	2	2	no	—
46	C₂×D₂₁₂	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×D₅₃	C₂³	1	2	yes	—
48	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
49	M₁₆×C₅₃	—	2	2	no	—
50	M₁₆×C₅₃	—	2	4	no	—
51	C₂×C₈×C₅₃	—	2	4	no	—
52	C₂×C₈×C₅₃	D₄	1	4	yes	complemented
53	C₂×C₈×C₅₃	C₂³	1	2	yes	—
54	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₅₃	C₂×C₄	1	4	yes	—
57	#11719921	C₂×C₄	1	4	yes	—
58	#11719919	C₂×C₄	1	4	yes	—
59	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄×Dic₅₃	C₂³	1	2	yes	—
62	#11719917	—	2	2	no	—
63	C₂×#88322	—	2	2	no	—
64	C₂×#88322	C₂³	1	2	yes	—
65	C₂×#88322	C₂³	1	2	yes	—
66	#11719912	—	2	4	no	—
67	C₈×D₅₃	—	2	4	no	—
68	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×D₅₃	C₂²×C₄	1	2	yes	—
71	Q₈×C₅₃	—	2	2	no	—
72	D₄×C₅₃	—	2	4	no	—
73	D₄×C₅₃	—	2	2	no	—
74	C₂×C₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
77	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×C₅₃	#41	1	2	yes	—
80	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
81	C₂²⋊D₅₃	—	2	4	no	—
82	C₂²⋊D₅₃	—	2	8	no	—
83	C₂×Dic₅₃	—	2	4	no	—
84	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
85	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
86	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
87	D₂₁₂	—	2	2	no	—
88	C₄×D₅₃	—	2	2	no	—
89	C₄×D₅₃	—	2	2	no	—
90	Dic₁₀₆	—	2	2	no	—
91	C₈×C₅₃	—	2	2	no	—
92	C₈×C₅₃	C₂×D₄	1	2	yes	—
93	C₈×C₅₃	C₂×D₄	1	2	yes	—
94	#88322	—	2	2	no	—
95	#88322	—	2	2	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	C₄×D₄	1	2	yes	—
100	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×D₅₃	—	2	2	no	—
102	C₂×D₅₃	—	2	4	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
106	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
107	Dic₅₃	—	2	2	no	—
108	Dic₅₃	—	2	2	no	—
109	Dic₅₃	#131	1	4	yes	complemented
110	#2931	—	53	1	no	maximal, Sylow
111	C₂×C₅₃	—	2	2	no	—
112	C₂×C₅₃	—	4	2	no	—
113	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
114	C₂×C₅₃	C₂×#131	1	2	yes	—
115	D₅₃	—	4	2	no	—
116	C₂×#131	—	53	1	no	—
117	C₂×#131	D₅₃	1	1	yes	characteristic, complemented
118	#532	—	53	1	no	—
119	#436	—	53	4	no	—
120	C₈×D₄	—	53	4	no	—
121	#423	—	53	1	no	—
122	#408	—	53	1	no	—
123	#408	—	53	1	no	—
124	C₂×C₄×C₈	—	53	1	no	—
125	C₅₃	#2931	1	1	yes	characteristic, complemented, Sylow
126	C₂×#41	—	53	1	no	—
127	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
128	#131	—	2	4	no	—
129	#131	—	106	4	no	—
130	C₂×M₁₆	—	53	1	no	—
131	C₂×M₁₆	—	53	2	no	—
132	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
133	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	53	2	no	—
136	C₂²×C₈	C₂×D₅₃	1	2	yes	—
137	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
138	C₄⋊Q₈	—	53	1	no	—
139	#127	—	53	1	no	—
140	#124	—	53	1	no	—
141	#124	—	53	1	no	—
142	#121	—	53	4	no	—
143	C₄×D₄	—	53	4	no	—
144	C₂×C₄²	—	53	1	no	—
145	C₄⋊C₈	—	53	4	no	—
146	C₂²⋊C₈	—	53	4	no	—
147	C₂²⋊C₈	—	53	4	no	—
148	C₄×C₈	—	53	2	no	—
149	C₄×C₈	—	53	2	no	—
150	#41	—	2	2	no	—
151	#41	—	2	2	no	—
152	#41	—	106	2	no	—
153	#41	—	106	2	no	—
154	C₂×Q₈	—	53	1	no	—
155	C₂×Q₈	C₄×D₅₃	1	1	yes	characteristic
156	C₂×D₄	—	53	4	no	—
157	C₂×D₄	—	53	2	no	—
158	C₂×D₄	—	53	1	no	—
159	C₂×D₄	C₄×D₅₃	1	2	yes	—
160	C₂×D₄	C₄×D₅₃	1	1	yes	characteristic
161	C₂²×C₄	—	53	2	no	—
162	C₂²×C₄	—	53	2	no	—
163	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
164	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
165	M₁₆	—	2	2	no	—
166	M₁₆	—	2	4	no	—
167	M₁₆	—	106	2	no	—
168	M₁₆	—	106	4	no	—
169	C₂×C₈	—	2	4	no	—
170	C₂×C₈	—	53	2	no	—
171	C₂×C₈	—	53	2	no	—
172	C₂×C₈	—	106	2	no	—
173	C₂×C₈	—	106	4	no	—
174	C₂×C₈	C₂²⋊D₅₃	1	4	yes	complemented
175	C₂×C₈	C₂²×D₅₃	1	2	yes	—
176	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
177	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
178	C₄⋊C₄	—	53	4	no	—
179	C₂²⋊C₄	—	53	4	no	—
180	C₂²⋊C₄	—	53	4	no	—
181	C₄²	—	53	1	no	—
182	C₄²	—	53	1	no	—
183	C₄²	—	53	2	no	—
184	C₂³	—	53	2	no	—
185	C₂³	C₂×C₄×D₅₃	1	2	yes	—
186	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
187	Q₈	—	2	2	no	—
188	Q₈	—	106	2	no	—
189	D₄	—	2	4	no	—
190	D₄	—	2	2	no	—
191	D₄	—	106	8	no	—
192	D₄	—	106	4	no	—
193	D₄	—	106	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	53	2	no	—
197	C₂×C₄	—	53	2	no	—
198	C₂×C₄	—	53	2	no	—
199	C₂×C₄	—	106	4	no	—
200	C₂×C₄	—	106	2	no	—
201	C₂×C₄	—	106	2	no	—
202	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
203	C₂×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
204	C₂×C₄	#11719945	1	2	yes	—
205	C₂×C₄	C₂×C₂²⋊D₅₃	1	2	yes	—
206	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	106	2	no	—
209	C₈	—	106	2	no	—
210	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
211	C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	106	2	no	—
216	C₂²	—	106	4	no	—
217	C₂²	C₄×C₂²⋊D₅₃	1	2	yes	—
218	C₂²	C₂²×C₄×D₅₃	1	1	yes	characteristic
219	C₄	—	2	2	no	—
220	C₄	—	53	4	no	—
221	C₄	—	106	2	no	—
222	C₄	—	106	2	no	—
223	C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
224	C₄	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄	C₂×#11719945	1	1	yes	characteristic
226	C₂	—	2	2	no	—
227	C₂	—	4	2	no	—
228	C₂	—	212	2	no	—
229	C₂	C₂×C₄×C₂²⋊D₅₃	1	1	yes	characteristic
230	C₂	#131×D₅₃	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂²×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#131×C₅₃	C₂
hypercenter	C₂×#131	D₅₃
hypocenter	C₅₃	#2931
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#131

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×#13133223	C₂	semidirect
3. #16810699	C₂	semidirect
4. C₈×C₂²⋊D₅₃	C₂	semidirect
5. #16810646	C₂	semidirect
6. #16810591	C₂	semidirect
7. #16810590	C₂	semidirect
8. C₂×C₈×Dic₅₃	C₂	semidirect
9. C₂×#131×C₅₃	C₂	semidirect
52. C₂×C₈×C₅₃	D₄	semidirect
109. Dic₅₃	#131	semidirect
117. C₂×#131	D₅₃	semidirect
125. C₅₃	#2931	semidirect
174. C₂×C₈	C₂²⋊D₅₃	semidirect